Is there any prohibition against giving a birthday card, (no present) written out before Shabbat to a person on Shabbat?
There is no problem with gift giving on shabbat.
Potential issues which could prevent the gift being given.... -the is no eruv in the community and the letter is in your home and you are on your way to give it to him at his home. obviously carrying would be the problem
-It is in a closed envelope which would require tearing to get to the letter